Showing papers on "Normal modal logic published in 2016"
TL;DR: Unified correspondence theory is extended to Kripke frames with impossible worlds and their associated regular modal logics and it is shown that additivity and multiplicativity turn out to be key to extend Jonsson’s original proof of canonicity to the full Sahlqvist class of certain regular distributives naturally generalizing distributive modal logic.
Abstract: We extend unified correspondence theory to Kripke frames with impossible worlds and their associated regular modal logics. These are logics the modal connectives of which are not required to be normal: only the weaker properties of additivity ◊x∨◊y=◊(x∨y) and multiplicativity □x∧□y=□(x∧y) are required. Conceptually, it has been argued that their lacking necessitation makes regular modal logics better suited than normal modal logics at the formalization of epistemic and deontic settings. From a technical viewpoint, regularity proves to be very natural and adequate for the treatment of algebraic canonicity Jonsson-style. Indeed, additivity and multiplicativity turn out to be key to extend Jonsson’s original proof of canonicity to the full Sahlqvist class of certain regular distributive modal logics naturally generalizing distributive modal logic. Most interestingly, additivity and multiplicativity are key to Jonsson-style canonicity also in the original (i.e. normal DML. Our contributions include: the definition of Sahlqvist inequalities for regular modal logics on a distributive lattice propositional base; the proof of their canonicity following Jonsson’s strategy; the adaptation of the algorithm ALBA to the setting of regular modal logics on two non-classical (distributive lattice and intuitionistic) bases; the proof that the adapted ALBA is guaranteed to succeed on a syntactically defined class which properly includes the Sahlqvist one; finally, the application of the previous results so as to obtain proofs, alternative to Kripke’s, of the strong completeness of Lemmon’s epistemic logics E2-E5 with respect to elementary classes of Kripke frames with impossible worlds.
55 citations
Posted Content•
TL;DR: In this article, a formal connection between correspondence phenomena, well known from the area of modal logic, and the theory of display calculi, originated by Belnap, is established.
Abstract: The present paper aims at establishing formal connections between correspondence phenomena, well known from the area of modal logic, and the theory of display calculi, originated by Belnap. These connections have been seminally observed and exploited by Marcus Kracht, in the context of his characterization of the modal axioms (which he calls primitive formulas) which can be effectively transformed into `analytic' structural rules of display calculi. In this context, a rule is `analytic' if adding it to a display calculus preserves Belnap's cut-elimination theorem. In recent years, the state-of-the-art in correspondence theory has been uniformly extended from classical modal logic to diverse families of nonclassical logics, ranging from (bi-)intuitionistic (modal) logics, linear, relevant and other substructural logics, to hybrid logics and mu-calculi. This generalization has given rise to a theory called unified correspondence, the most important technical tools of which are the algorithm ALBA, and the syntactic characterization of Sahlqvist-type classes of formulas and inequalities which is uniform in the setting of normal DLE-logics (logics the algebraic semantics of which is based on bounded distributive lattices). We apply unified correspondence theory, with its tools and insights, to extend Kracht's results and prove his claims in the setting of DLE-logics. The results of the present paper characterize the space of properly displayable DLE-logics.
40 citations
28 Jun 2016
TL;DR: The modal independence logic MIL as discussed by the authors, a modal logic that can explicitly talk about independence among propositional variables, is a variant of Vaananen's modal dependence logic MDL.
Abstract: This article introduces modal independence logic MIL, a modal logic that can explicitly talk about independence among propositional variables. Formulas of MIL are not evaluated in worlds but in sets of worlds, so called teams. In this vein, MIL can be seen as a variant of Vaananen’s modal dependence logic MDL. We show that MIL embeds MDL and is strictly more expressive. However, on singleton teams, MIL is shown to be not more expressive than usual modal logic, but MIL is exponentially more succinct. Making use of a new form of bisimulation, we extend these expressivity results to modal logics extended by various generalized dependence atoms. We demonstrate the expressive power of MIL by giving a specification of the anonymity requirement of the dining cryptographers protocol in MIL. We also study complexity issues of MIL and show that, though it is more expressive, its satisfiability and model checking problem have the same complexity as for MDL.
38 citations
TL;DR: Stable canonical rules are introduced and it is proved that each normal modal multi-conclusion consequence relation is axiomatizable by stable canonical rules.
Abstract: We introduce stable canonical rules and prove that each normal modal multi-conclusion consequence relation is axiomatizable by stable canonical rules. We apply these results to construct finite refutation patterns for modal formulas, and prove that each normal modal logic is axiomatizable by stable canonical rules. We also define stable multi-conclusion consequence relations and stable logics and prove that these systems have the finite model property. We conclude the paper with a number of examples of stable and nonstable systems, and show how to axiomatize them.
32 citations
TL;DR: Very general, though partly non-constructive, methods that cover all previous examples, and extend to an infinite family of modal logics are provided.
30 citations
01 Jan 2016
TL;DR: The natural deduction hybrid systems and modal logics is universally compatible with any devices to read and an online access to it is set as public so you can get it instantly.
Abstract: natural deduction hybrid systems and modal logics is available in our book collection an online access to it is set as public so you can get it instantly. Our book servers hosts in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Merely said, the natural deduction hybrid systems and modal logics is universally compatible with any devices to read.
26 citations
TL;DR: Two conjectures on the covering-based rough set semantics for modal logics in 35 are answered: the C 2 and C 5 semantics give rise to the same modal system S4.
22 citations
TL;DR: A process calculus with an explicit representation of resources in which processes and resources co-evolve is described, and the resource semantics is formulated in such a way that soundness and completeness of bisimulation are obtained with respect to logical equivalence for the naturally full range of logical connectives and modalities.
22 citations
TL;DR: This paper defines a full and faithful translation of the basic modal logic K over possibility models into a bimodal logic of partial functions over partial orders, and shows how to modulate this analysis by varying across logics and model classes that have independent topological motivations.
Abstract: In this paper we develop a bimodal perspective on possibility semantics, a framework allowing partiality of states that provides an alternative modeling for classical propositional and modal logics [Humberstone, 1981, Holliday, 2015]. In particular, we define a full and faithful translation of the basic modal logic K over possibility models into a bimodal logic of partial functions over partial orders, and we show how to modulate this analysis by varying across logics and model classes that have independent topological motivations. This relates the two realms under comparison both semantically and syntactically at the level of derivations. Moreover, our analysis clarifies the interplay between the complexity of translations and axiomatizations of the corresponding logics: adding axioms to the target bimodal logic simplifies translations, or vice versa, complex translations can simplify frame conditions. We also investigate a transfer of first-order correspondence theory between possibility semantics and its bimodal counterpart. Finally, we discuss the conceptual trade-o between giving translations and giving new semantics for logical systems, and we identify a number of further research directions to which our analysis gives rise.
20 citations
Proceedings Article•
30 Aug 2016TL;DR: A general framework is proposed, based on a focused version of the labeled sequent calculus by Negri, augmented with some parametric devices allowing to restrict the set of proofs, to emulate in this framework the behavior of a range of existing formalisms and proof systems for modal logic.
Abstract: Several deductive formalisms (e.g., sequent, nested sequent, labeled sequent, hyperse-quent calculi) have been used in the literature for the treatment of modal logics, and some connections between these formalisms are already known. Here we propose a general framework, which is based on a focused version of the labeled sequent calculus by Negri, augmented with some parametric devices allowing to restrict the set of proofs. By properly defining such restrictions and by choosing an appropriate polarization of formulas, one can obtain different, concrete proof systems for the modal logic K and for its extensions by means of geometric axioms. In particular, we show how to use the expressiveness of the labeled approach and the control mechanisms of focusing in order to emulate in our framework the behavior of a range of existing formalisms and proof systems for modal logic.
TL;DR: In this paper, a systematic study of the lattice of logics containing BK was carried out, where the classes (or rather sublattices) of so-called explosive, complete and classical Belnapian modal logics were introduced.
Abstract: Let K be the least normal modal logic and BK its Belnapian version, which enriches K with ‘strong negation’. We carry out a systematic study of the lattice of logics containing BK based on: • introducing the classes (or rather sublattices) of so-called explosive, complete and classical Belnapian modal logics; • assigning to every normal modal logic three special conservative extensions in these classes; • associating with every Belnapian modal logic its explosive, complete and classical counterparts. We investigate the relationships between special extensions and counterparts, provide certain handy characterisations and suggest a useful decomposition of the lattice of logics containing BK.
TL;DR: A sound and complete axiomatization of these logics with respect to the two equivalent Kripke semantics and type spaces semantics is provided and a probabilistic extension of the result is presented.
Abstract: In the literature there are at least two main formal structures to deal with situations of interactive epistemology: Kripke models and type spaces. As shown in many papers (see Aumann and Brandenburger in Econometrica 36:1161–1180, 1995; Baltag et al. in Synthese 169:301–333, 2009; Battigalli and Bonanno in Res Econ 53(2):149–225, 1999; Battigalli and Siniscalchi in J Econ Theory 106:356–391, 2002; Klein and Pacuit in Stud Log 102:297–319, 2014; Lorini in J Philos Log 42(6):863–904, 2013), both these frameworks can be used to express epistemic conditions for solution concepts in game theory. The main result of this paper is a formal comparison between the two and a statement of semantic equivalence with respect to two different logical systems: a doxastic logic for belief and an epistemic–doxastic logic for belief and knowledge. Moreover, a sound and complete axiomatization of these logics with respect to the two equivalent Kripke semantics and type spaces semantics is provided. Finally, a probabilistic extension of the result is also presented. A further result of the paper is a study of the relationship between the epistemic–doxastic logic for belief and knowledge and the logic STIT (the logic of “seeing to it that”) by Belnap and colleagues (Facing the future: agents and choices in our indeterminist world, 2001).
TL;DR: This work investigates hypersequent calculi given by rules of the newly introduced format of hypersequent rules with context restrictions used to prove uniform syntactic cut elimination, decidability and complexity results and introduces transformations between hypersequentrules and Hilbert axioms.
Proceedings Article•
01 Apr 2016TL;DR: It is shown that the conditional "knowing value" logic proposed by Wang and Fan can be viewed as a disguised normal modal logic by treating the negation of the Kv operator as a special diamond, and the resulting logic with the binary diamond has a transparentnormal modal system, which sharpens the understanding of the "knowed value" Logic and simplifies some previously hard problems.
Abstract: Recent years witness a growing interest in nonstandard epistemic logics of "knowing whether", "knowing what", "knowing how", and so on. These logics are usually not normal, i.e., the standard axioms and reasoning rules for modal logic may be invalid. In this paper, we show that the conditional "knowing value" logic proposed by Wang and Fan \cite{WF13} can be viewed as a disguised normal modal logic by treating the negation of the Kv operator as a special diamond. Under this perspective, it turns out that the original first-order Kripke semantics can be greatly simplified by introducing a ternary relation $R_i^c$ in standard Kripke models, which associates one world with two $i$-accessible worlds that do not agree on the value of constant $c$. Under intuitive constraints, the modal logic based on such Kripke models is exactly the one studied by Wang and Fan (2013,2014}. Moreover, there is a very natural binary generalization of the "knowing value" diamond, which, surprisingly, does not increase the expressive power of the logic. The resulting logic with the binary diamond has a transparent normal modal system, which sharpens our understanding of the "knowing value" logic and simplifies some previously hard problems.
01 Sep 2016
TL;DR: A sound and complete sequent calculus is presented for a modal fragment of XPath equipped with data (in)equality tests over finite data trees that yields the optimal PSPACE complexity bound for its validity problem.
Abstract: We investigate the proof theory of a modal fragment of XPath equipped with data (in)equality tests over finite data trees, i.e., over finite unranked trees where nodes are labelled with both a symbol from a finite alphabet and a single data value from an infinite domain. We present a sound and complete sequent calculus for this logic, which yields the optimal PSPACE complexity bound for its validity problem.
Posted Content•
TL;DR: This is a survey of {\lambda}-calculi that, through the Curry-Howard isomorphism, correspond to constructive modal logics that took place in the 1990s and early 2000s.
Abstract: This is a survey of {\lambda}-calculi that, through the Curry-Howard isomorphism, correspond to constructive modal logics. We cover the prehistory of the subject and then concentrate on the developments that took place in the 1990s and early 2000s. We discuss logical aspects, modal {\lambda}-calculi, and their categorical semantics. The logics underlying the calculi surveyed are constructive versions of K, K4, S4, and LTL.
TL;DR: A bi-intuitionistic modal logic, called BISKT, with two adjoint pairs of tense operators, with decidability, complexity and correspondence results is introduced, and it is shown the logic has the effective finite model property.
TL;DR: It is established that the absence of a distributivity assumption of conjunctions over disjunctions and conversely has no effect on the interpretation of boxes and diamonds, which are interpreted exactly as in classical normal modal logics.
Abstract: A notorious difficulty with modal extensions over a non-distributive propositional basis is to construct canonical Kripke models (time flow structures, when a temporal interpretation is intended) that respect the well-established intuitions about the meaning of modal (temporal) operators. Indeed, advances in modal logics over a non-distributive propositional basis over the last decade or so address a number of issues of significance, such as Sahlqvist (algorithmic) correspondence and completeness, yet they do this while resting on a notion of frame and model, canonical or otherwise, that compromises the intuitive semantics of the modal operators in important ways. This becomes particularly apparent when a dynamic, or a temporal reading of boxes and diamonds is intended. This article is restricted to the simplest temporal logic, a Priorean Tense Logic over a negation and implication free non-distributive propositional basis. We build up to this system by considering separately systems with modal operators in isolation, or in related groups, and we prove, for each system, completeness via a traditional canonicity argument. We establish that the absence of a distributivity assumption of conjunctions over disjunctions and conversely has no effect on the interpretation of boxes and diamonds, which are interpreted exactly as in classical normal modal logics. The motivation of this work resides in the realization that at least part of the reason for the manifest dissatisfaction with studying and applying modal logics over a non-distributive propositional basis is due precisely to the fact that a large number of semantic intuitions, results and techniques familiar from distributive logics seem to have to be inescapably abandoned. We argue in this article that this is not necessarily the case. The results we present can be applied, e.g. in studying dynamic, or temporal extensions of Orthomodular Quantum Logic, as well as in the relational semantics for Substructural Logics.
TL;DR: In this article, a class of modal logics rendered insensitive to reflexivity by way of a modification to the semantic definition of the modal operator is analyzed. And a sufficient semantic condition describing when a similarly general soundness result is also available.
Abstract: We analyze a class of modal logics rendered insensitive to reflexivity by way of a modification to the semantic definition of the modal operator. We explore the extent to which these logics can be characterized, and prove a general completeness theorem on the basis of a translation between normal modal logics and their reflexive-insensitive counterparts. Lastly, we provide a sufficient semantic condition describing when a similarly general soundness result is also available.
Posted Content•
TL;DR: This paper provides a call-by-name and aCall- by-value term calculus, both of which have a Curry-Howard correspondence to the box fragment of the intuitionistic modal logic IK and show the strong normalizability and the confluency of the calculi.
Abstract: This paper provides a call-by-name and a call-by-value term calculus, both of which have a Curry-Howard correspondence to the box fragment of the intuitionistic modal logic IK. The strong normalizability and the confluency of the calculi are shown. Moreover, we define a CPS transformation from the call-by-value calculus to the call-by-name calculus, and show its soundness and completeness.
TL;DR: So-called epistemic Kripke structures for intuitionistic (common) knowledge/belief are defined and corresponding deductive systems are introduced that are sound and complete with respect to these semantics.
TL;DR: It is shown that the intuitionistic consequence system is embeddable in the modal (S4) one by the usual modal translation prefixing □ to every subformula of the translated formula.
Abstract: Abstract We construct four binary consequence systems axiomatizing entailment relations between formulas of classical, intuitionistic, dual-intuitionistic and modal (S4) logics, respectively. It is shown that the intuitionistic consequence system is embeddable in the modal (S4) one by the usual modal translation prefixing □ to every subformula of the translated formula. An analogous modal translation of dual-intuitionistic formulas then consists of prefixing ◊ to every subformula of the translated formula. The philosophical importance of this result is briefly discussed.
TL;DR: It is shown that S1+SP is strictly contained between S1 and S3 and differs from S2, the weakest modal logic containing S1 such that strict equivalence is axiomatized by propositional identity.
Abstract: The modal systems S1--S3 were introduced by C. I. Lewis as logics for strict implication. While there are Kripke semantics for S2 and S3, there is no known natural semantics for S1. We extend S1 by a Substitution Principle SP which generalizes a reference rule of S1. In system S1+SP, the relation of strict equivalence $\varphi\equiv\psi$ satisfies the identity axioms of R. Suszko's non-Fregean logic adapted to the language of modal logic (we call these axioms the axioms of propositional identity). This enables us to develop a framework of algebraic semantics which captures S1+SP as well as the Lewis systems S3--S5. So from the viewpoint of algebraic semantics, S1+SP turns out to be an interesting modal logic. We show that S1+SP is strictly contained between S1 and S3 and differs from S2. It is the weakest modal logic containing S1 such that strict equivalence is axiomatized by propositional identity.
TL;DR: A fibrational categorical semantics for the modal necessity-only fragment of constructive modal type theory, both with and without dependent types, is described, which provides evidence that the type theory is meaningful.
01 Jan 2016
TL;DR: An equivalence between deterministic modal transition systems and a language-based formalism called modal specifications is developed, and the claim follows from a known equivalences between the conjunctive nu-calculus andmodal specifications.
Abstract: In this paper, the synthesis of bounded Petri nets from deterministic modal transition systems is shown to be undecidable. The proof is built from three components. First, it is shown that the problem of synthesising bounded Petri nets satisfying a given formula of the conjunctive nu-calculus (a suitable fragment of the mu-calculus) is undecidable. Then, an equivalence between deterministic modal transition systems and a language-based formalism called modal specifications is developed. Finally, the claim follows from a known equivalence between the conjunctive nu-calculus and modal specifications.
Proceedings Article•
12 Feb 2016TL;DR: A multi-modal version of Second-order Propositional Modal Logic (SOPML), an extension of modal logic with propositional quantification, is introduced and its usefulness as a specification language for knowledge representation as well as temporal and spatial reasoning is illustrated.
Abstract: This paper is aimed as a contribution to the use of formal modal languages in Artificial Intelligence. We introduce a multi-modal version of Second-order Propositional Modal Logic (SOPML), an extension of modal logic with propositional quantification, and illustrate its usefulness as a specification language for knowledge representation as well as temporal and spatial reasoning. Then, we define novel notions of (bi)simulation and prove that these preserve the interpretation of SOPML formulas. Finally, we apply these results to assess the expressive power of SOPML.
TL;DR: This work translates unconstrained and constrained input/output logics as introduced by Makinson and van der Torre to modal logics, using adaptive logics for the constrained case, and obtains a proof-theoretic (dynamic) characterization of input/ Output logics.
Abstract: We translate unconstrained and constrained input/output logics as introduced by Makinson and van der Torre to modal logics, using adaptive logics for the constrained case. The resulting reformulation has some additional benefits. First, we obtain a proof-theoretic (dynamic) characterization of input/output logics. Second, we demonstrate that our framework naturally gives rise to useful variants and allows to express important notions that go beyond the expressive means of input/output logics, such as violations and sanctions.